We present a numerical approximation of Darcy's flow through a fractured porous medium which employs discontinuous Galerkin methods on polytopic grids. For simplicity, we analyze the case of a single fracture represented by a (d-1)-dimensional interface between two d-dimensional subdomains, d = 2, 3. We propose a discontinuous Galerkin finite element approximation for the flow in the porous matrix which is coupled with a conforming finite element scheme for the flow in the fracture. Suitable (physically consistent) coupling conditions complete the model. We theoretically analyze the resulting formulation, prove its well-posedness, and derive optimal a priori error estimates in a suitable (mesh-dependent) energy norm. Two-dimensional numerical experiments are reported to assess the theoretical results.
We present a comprehensive review of the current development of PolyDG methods for geophysical applications, addressing as paradigmatic applications the numerical modeling of seismic wave propagation and fracture reservoir simulations. We first recall the theoretical background of the analysis of PolyDG methods and discuss the issue of its efficient implementation on polytopic meshes. We address in detail the issue of numerical quadrature and recall the new quadrature free algorithm for the numerical evaluation of the integrals required to assemble the mass and stiffness matrices introduced in [22]. Then we present PolyDG methods for the approximate solution of the elastodynamics equations on computational meshes consisting of polytopic elements. We review the well-posedness of the numerical formulation and hp-version a priori stability and error estimates for the semi-discrete scheme, following [10]. The computational performance of the fully-discrete approximation obtained based on employing the PolyDG method for the space discretization coupled with the leap-frog time marching scheme are demonstrated through numerical experiments. Next, we address the problem of modeling the flow in a fractured porous medium and we review the unified construction and analysis of PolyDG methods following [16]. We show, in a unified setting, the well-posedness of the numerical formulations and hp-version a priori error bounds, that are then validated through numerical tests. We also briefly discuss the extendability of our approach to handle networks of partially immersed fractures and networks of intersecting fractures, recently proposed in [15].
We propose a formulation based on discontinuous Galerkin methods on polygonal/polyhedral grids for the simulation of flows in fractured porous media. We adopt a model for single-phase flows where the fracture is modelled as a (d − 1)-dimensional interface in a d-dimensional bulk domain and the flow is governed by the Darcy's law in both the bulk and the fracture. The two problems are then coupled through physically consistent conditions. We focus on the numerical approximation of the coupled bulk-fracture problem, discretizing the bulk problem in mixed form and the fracture problem in primal form. We present an priori h-and p-version error estimate in a suitable (mesh-dependent) energy norm and numerical tests assessing it.
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